Generalizations to Complete Demand Systems
(1) Socio-Demographic Variables
The treatment of demographic effects in the context of demand systems dates from Barten (1964) and more recently from Parks and Barten; Lau,
Lin, and Yotopoulos; Muellbauer; and Pollak and Wales (1978, 1980,
1981). Pollak and Wales (1981) describe for general procedures for incorporating demographic variables into demand systems:
(1) Demographic Translating
(2) Demographic Scaling
(3) Gorman and Reverse Gorman Specifications
(4) The Modified Prais-Houthakker Procedure
Other ways of handling the problem of incorporating these types of variables – a set of separate demand relationships can be estimated for each of the demographic variables of interest. From a practical viewpoint, this method is usually not possible because of data limitations.
Further, the demographic variables may be incorporated into the utility function and thus will appear in the equations of the complete demand system.
169

Advanced by Lau, Lin, and Yotopoulos, and Parks and Barten, stipulates embedding these variables, either continuous or discrete, into the direct or indirect utility function.
U

U ( q
1
,..., q n
, a
1
,..., a r
) OR
  
( p
1
,..., p n
, y , a
1
,..., a r
)
Each a i
, i = 1, …, r reflects one dimension of the r household characteristics.
The demand functions which arise then depend on the vector of socioeconomic and demographic factors, (a
1
, …, a r
). In short, these researchers specify a complete utility maximization model for the consumption behavior of households which takes into account the effects of differential composition of the households in a general way.
(2) Dynamic Complete Demand System
--Use of a state adjustment model in which quantities purchased depend on existing stocks of either physical stocks of goods or psychological stocks of habits
--Use a dynamic utility function (Phlips)
--Cast the problem into a control theory framework in which the consumer is attempting to maximize a discounted utility function subject to wealth and stocks constraints.
Application of state adjustment model –Green, Hassan, Johnson; Houthakker,
Taylor
170

Usefulness of Complete Demand Systems
(1) The generation of a massive volume of empirical results consistent with the theory of consumer behavior
(2) Information to test hypotheses about postulates & restrictions directly obtainable from the theory
The voluminous results depend on the specific functional forms and the particular aggregate consumption categories and specific individual goods being analyzed.
Complete demand systems usually generate estimates of all own-price elasticities, marginal budget shares of all aggregate consumption categories and individual goods, and all Slutsky or income compensated own-price and cross-price elasticities. In addition to the above parameter estimates, the additive complete demand systems provide estimates of welfare indicators such as income flexibilities and the marginal utility of money. The extended linear expenditure system (ELES) as developed by
Lluch, yields an estimate of the marginal propensity to consume and in this sense links consumer demand theory with macroeconomic theory.
Some system specifications also provide estimates of elasticities of substitution among goods and of subsistence levels of consumption or expenditure for individual goods.
171

Generalized complete demand systems which include socioeconomic or demographic variable also generate coefficients or elasticities for assessing the impacts of these variables on quantities demanded of individual goods and aggregate consumption categories. Dynamic complete demand systems provide estimates of the impacts of habit formation and of the path of adjustment of consumers to changes in prices, total expenditure and other variables.
The use of complete demand systems has provided information for testing postulates theoretically derived from the economic theory of consumer satisfaction subject to a budget constraint. The simplest of these have entailed hypotheses about the directional impacts of changes in prices and total expenditures on quantities demanded of individual goods. Complete demand systems have also enabled testing hypotheses about the validity of selected restrictions such as homogeneity constraint, symmetry restrictions, and adding-up constraints which are derived directly from the economic theory of consumer behavior. In addition, the use of complete demand systems has permitted testing hypotheses about various grouping of goods according to alternative separability rules. Brown and
Deaton point out that the economic theory of consumer behavior in conjunction with complete demand systems has provided economists with an experience of testing and applying a sophisticated set of theoretically derived postulates to actual data which is an opportunity not
172 often available to social scientists.

Data, Model Specification, and Estimation
First, complete demand systems, even those with a few highly aggregated consumption categories have a large number of parameters to be estimated. From a practical viewpoint interest is in individual goods of at least some consumption categories which almost always assures estimation of a large number of parameters. Even though various restrictions may be used to reduce the number of parameters to be estimated, estimation of the parameters of complete demand systems requires a rather substantial sample size. This necessity of a large sample size is most restrictive in terms of the use of time series data. A long enough time series may not be available to estimate the parameters of a complete demand system with consumption categories sufficiently disaggregated for the model to be useful, or if the data are available, it covers such a long time span that substantial structural changes may have occurred during the time period.
The second data issue occurs when survey or cross-sectional data on individual consuming units are being used. This issue is not unique to the estimation of complete demand systems. It effects the estimation of any type of demand relationship. The issue is concerned with the use of zero quantity observations. The question is should these observations be deleted or should they be included?
173

Because of a need to reduce the number of parameters to be estimated or because parameters of individual goods of a set of consumption categories are not of interest, it may be desirable or necessary to group goods together into categories.
In other words, the goods are grouped into separable groups, and goods in each separable group tend to interact closely while goods between separable groups do not. Furthermore, if the parameters of the individual goods of a particular group are not of interest, that group may be handled as an aggregate.
Developing the appropriate weights for individual goods in an aggregate and developing an appropriate price to represent the aggregate usually pose some difficult measurement problems.
174

There are various kinds of separability with guides for allocating individual goods to the different groups. Unfortunately, the guides for allocating the individual goods into groups are based on changes in the ratio of the marginal utilities (marginal rate of substitution) of two goods with respect to a change in quantity of a third good.
Information about the change in the ratio of the marginal utilities of pairs of goods is not available and the guides are not very useful in empirical analysis. Furthermore, results of statistical procedures, such as cluster analysis factor analysis, are something less than spectacular and usually result in an indeterminate situation for several goods. In most empirical analyses involving separability, the individual goods are grouped according to similar physical characteristics and ability to yield satisfaction as judged by the analyst. Different analysts have different perspectives about various individual goods and perceive the satisfaction yielding attributes of goods differently and, thus, the grouping of goods to separable groups is arbitrary to some degree.
175

Practitioners are continually faced with theoretically satisfactory models imbued with practical difficulties; estimation methods should take into account parameter nonlinearity, cross-equation correlation, variance-covariance singularity of the disturbance terms, and various parameter restrictions.
Appropriate estimation approaches maximum likelihood IZEF
Computational burden is not to be underestimated
176

The large number of coefficients in complete demand systems seemingly is an ubiquitous issue in estimation. A relatively large sample size is required. Theoretical restrictions, such as symmetry, homogeneity, and Engel aggregation help reduce the number of parameters which have to be estimated.
For complete demand systems derived from the maximization of a constrained utility function, these restrictions are automatically satisfied.
177

Aggregation is unavoidable in estimation of complete demand systems and, thus, analyses are subject to aggregation problems. In particular, difficulties of grouping homogeneous goods, obtaining weights for individual goods in an aggregate, and obtaining price and quantity measures of aggregates are common problems. Also, there is the difficulty of interpreting the estimated coefficients of aggregate goods. Furthermore, in the aggregation over consumers, the assumption that prices are predetermined may be called into question. Consequently, the supply relationship may need to be considered explicitly in order to handle the simultaneity between prices and quantities.
178

Serial Correlation can pose some real serious problems in the estimation of complete demand systems. The inefficiency in a statistical sense which accompanies serial correlation is undesirable, but corrections can be made for its effects.
However, the most serious impacts may occur in the complete demand systems derived from maximization of a utility function subject to a budget constraint. Estimation of parameters of these systems requires deletion of an individual good or an aggregate good and the estimates of the parameters of the deleted good are derived algebraically. Existence of serial correlation in expenditure systems results in the numerical values of the estimated coefficients depending on the deleted good.
179

Multicollinearity may be a serious problem in estimating complete demand systems particularly if time-series data are being used. Each of the demand equations encompasses the prices of all goods being considered and the total expenditure.
Prices of substitutes and complements tend to be correlated over time and of course the larger the system, the greater the number of price variables entering each equation.
Aggregation, separability, and the use of theoretical restrictions all may help reduce multicollinearity. In some specifications other procedures, including ridge regression techniques or modifications of ridge regression techniques, may be helpful.
180

rity Some useful specifications of complete demand systems encompass equations which are nonlinear in parameters. The estimation task and cost of estimation are considerably greater than for linear systems. Software packages for nonlinear estimation of systems of equations are not as prevalent but there are a few available.
Hopefully, vast improvements will occur in nonlinear estimation techniques in the near future to reduce estimation tasks and costs.
181

Complete demand systems are an attractive approach to the analyses of consumer demand behavior. Complete demand systems generate a large number of parameters consistent with economic theory and provide a quantitative basis for assessing consumer responses to a large number of economic variables.
Of course, estimation of parameters of the complete demand systems are not without problems. But data bases, estimation techniques, and computers are becoming available which are making the complete demand systems approach an even more feasible tool of analysis. Hopefully, in the near future, further progress will be made in incorporating non-economic variables into complete demand systems, applications of nonlinear estimation, and in the formulation and estimation of linear and nonlinear dynamic complete demand systems
182

Will consider several widely used empirical formulations
Linear Expenditures Systems (LES) utility function approach
Rotterdam Model directly specified demand system
Almost Ideal Demand System (AIDS) flexible functional form
State Adjustment Model dynamic demand system
Translog Models
183

Surveys
Brown & Deaton (1972)
Barten (1977)
Deaton (1986)
184

Demand Models fall into 1 of 3 broad categories:
--derived from a specific optimization problem
--based on an approximation to an unspecified system
--arbitrarily specified
185

--Stone’s Linear
Expenditure System (1954)
Houthakker’s Direct
Addilog Model (1960)
Direct
Utility
Function
--Leser (1941)
Houthakker’s Direct
Addilog Model (1960)
Indirect
Utility
Function
186

--Theil’s Rotterdam Model (1965)
--Christensen, et al. (1975)
Jorgenson and Lau’s 1975
Translog Model
--Deaton & Muellbauer’s Almost Ideal Demand System (1980)
--Constant elasticity of demand system
--Generalized Addilog Model (Bewley 1982) developed from Theil’s (1969) multinomial extension of the linear logit model
187

Models differ due to:
1.
Different “explicit” assumptions underlying the demand functions; e.g., want independence vs. separability.
2.
Different “implicit” assumptions due to a particular functional form for the demand functions and/or utility function.
3.
Unique characteristics of the data used to test and estimate the demand parameters
Linear Expenditure System
The linear expenditure system has been, perhaps, the most used empirical demand formulation and is the underlying model for Stone’s consumer expenditure studies.
188

Application – Stone pp. 517-519
Commodity groups:
1. meat, fish, dairy products & fats
2. fruit and vegetables
3. drink and tobacco
4. household variable expenses
5. durable goods
6. all other goods and services
189

Linear Expenditure System
Demand functions derived from utility maximization the model of consumer demand which results when the utility function is of the (Stone-Geary)
Klein-Rubin form
U
 i n 

1
B i ln( q i
  i
)
0

B i

1 ,

B i

1 , q i
  i

0
Directly additive – not appropriate when a fine classification of goods is involved p i q i
 p i
 i

B i
(y -
 p j
 j
)
Linear Engel Curves approximate proportionality between own-price & expenditure elasticities
Interpretation (attributable to Samuelson)
 i  if positive represent “minimum requirements” subsistence quantities
190

Given this interpretation, the expenditure on the ith commodity consists of the expenditure on the minimum required quantity of the ith commodity plus the proportion of the budget leftover after the expenditure on ALL minimum requirements is accounted for. This proportion, B i
, is the marginal budget share, and the dollar amount to be allocated is called
“supernumerary” income.
Own-Price Elasticity (E ii
)

1

( 1

B i
)
 i q i
Income Elasticity
B i
W i
Cross-Price Elasticity (E ij
)

B i
P j
 j p i q i 191

192

193

194

195

196

The Rotterdam Model
Introduction – one aspect of empirical demand studies is to validate theory by subjecting theoretical constraints to statistical tests using observable data.
The usual constraints of homogeneity, additivity, and symmetry are normally expressed in elasticity terms and as such may be difficult to test directly.
The Rotterdam model developed by Theil and Barten is essentialy the formulation of a demand system in a format which will easily lend itself to empirical verification of the theoretical demand constraints.
Appeal of the model is that:
(a) one does not have to assume a particular utility function
(b) allows one to investigate all postulates and implications of theory.
Disadvantage is that it does not satisfy integrability conditions which are necessary to derive the fundamental demand equations (see Brown &
Deaton, pp. 1160, 1161, 1164)
197

Despite this theoretical shortcoming, the model has been used extensively to test the theoretical constraints.
While the symmetry constraints cannot usually be rejected in empirical applications, the homogeneity condition is rejected when commodities are fairly disaggregate. This finding implies the existence not the absence of money illusion, but one cannot reject theory without more research into some of the fallabilities of the model to conform to all the theoretical requisites.
198

Rotterdam Model
Theirl (1965) – Barten (1964-66) work in differentials & first order approximations to the demand functions d log q i
  i d log y


 j ij d log p j
Use Slutsky Decomposition,
 ij
 * ij
  i w j
Where: d log q i
  i
( d log y

 k w k d log p k
)

 j
 * ij d log p j d log q i
  i d log Y


 j
* ij d log p j d log y
 d log y

 k w k d log p k
199

NOTE: d log y

 k w k d log y
ALSO: d log y

 k w k d log q k d log q i
  i d log y


 j
* ij d log p j w i d log q i
 w i
 i d log y

 j w i
 * ij d log p j w i
 i represents the marginal propensity to spend on the ith good
Slutsky symmetry restriction
 d log y w i
 * ij
 w j
 * ji represents the proportional change in real total expenditure
200

Let b i c ij

 w i w i
 i
 * ij
 w i d log q i
 b i d log y

 j c ij d log p j
For symmetry condition to hold, c ij
= c ji
This equation can be estimated provided the differentials are replaced by finite approximations and provided we are prepared to treat the b i c ij
’s as constant parameters.
c ij
 w i

Homogeneity
* j ij n 

1
 c ij


 p i
0 y q i



 k ij q i p j




 p i y p j
’s and

 k ij w
* it
Dq it
 b i



Dy t

 k w
* kt
Dp kt




 j c ij
Dp jt
Where:
Dq it
= log[q it
/ q it-1
]
Dp jt
= log[p jt
/ p jt-1
]
Dy t w it
*
= log[y t
= ½(w it
/ y t-1
]
+ w it-1
)
201

Adding-up Restrictions

 k b k
1 ,
 k c kj

0
Homogeneity restrictions
 jk

0 k c
Slutsky symmetry c ij
 c ji
Since all prices are positive, c is negative semi-definite

K = [k ij
] is negative semi-definite. Consequently, symmetry & negativity of K can be tested using the same restrictions on c.
Singularity of c matrix due to adding-up and homogeneity restrictions
Also, the constraints of separability and additivity may be written in terms of relationships between c ij additivity, c ij
   b i b and b i
. In particular, under the assumption of j
i
 j
202

The first tests of homogeneity and symmetry were carried out by
Barten (1967 & 1969) using Dutch data covering both pre war and post war experience. See also Deaton (1974).
Homogeneity of demands is in clear conflict with the evidence, while symmetry as an additional restriction is not particularly damaging.
Perhaps the most interesting feature of the Rotterdam model is its ability to model the whole substitution matrix.
Barten & Geyskens (1975) have actually enforced the negativity restrictions in the estimation of demand systems and have not in general found any great conflict with the evidence.
203

The Almost Ideal Demand System
-introduced by Deaton & Muellbauer (1980)
--possesses a functional form consistent with household budget data
--extension of model proposed by Working & Leser (1943)
AIDS model derived by use of duality concepts log c

( u
0
, p )
 
0

 k
 k log p k

1 / 2
 k j
 kj log p k log p j
 uB
0
 k represents outlay required for a minimal standard of living when P i for all i p k
B k
= 1
Apply Shepherd’s Lemma
Alternatively,
C ( u , p )
 exp[ a ( p )
 ub ( p )] a ( p )

 i
 i ln p i

1 / 2
 i j
 ij ln p i ln p j b ( p )

B
0 n
 i p i
Bi
204

Working (1943) Leser (1963) w i
  i

B i log y
 w i
  i

 j
 ij log p j

B i log( y / P ) log P
 
0

 k
 k log p k

1 / 2
 k
 j
 kj log p k log p j
Stone’s Approximation log P
*   k w k log p k
P

P
*
LA/AIDS w i
  i
* 

 ij log p j

B i log( y / P
*
)
 i
*   i

B i log

205

Adding-Up
 i
 i

1 ;
 i
 ij

0 ;
 i
B i

0
Homogeneity
 j
 ij

0
Symmetry
 ij
  ji
206

The impacts of changes in relative prices on the budget shares work through the terms
 ij
. Each
 ij represents 100 times the effect on the ith budget share of a 1 percent change in the jth price with Y/P held constant. Changes in real expenditure operate through the B i coefficients.
Elasticity Calculations
 i

1

B i
/ W i
 ii
 
1

1 w i



 ii

B i



 i
 n  r
 ri ln p r








 ij

1 w i



 ij

B i



 j
 n  r
 rj ln p r






See Green & Alston (1990)
207

The negativity conditions are satisfied if the matrix C defined by
C ij
=
 ij
+ B i
B j log (y/p) – W i
 ij
+ W i
W j is negative semi-definite.
Approximate log P by log P
  w k log p k
In many practical situations where prices are relatively collinear, P will thus be approximately proportional to any appropriately defined index. Such an index can be calculated directly before estimation – consequently we get an estimation problem which is relatively straight forward, a situation in sharp contrast to the estimation of the translog models.
B i determines whether goods are luxuries or necessities with B i increases with y so that good i is a luxury.
> 0, w i
B i
< 0
 necessities
 Engel’s Law
The
 ij
’s measure the change in the ith budget share following a unit proportional change in p j
Rejection of homogeneity using postwar British annual data 1954-74.
208

SHAZAM PROGRAM for Rotterdam Model
/ restrict System 3 5
OLS w
1
DQ1 AY DP1 DP2 DP3 DP4
OLS w
2
DQ2 AY DP1 DP2 DP3 DP4
OLS w
3
DQ3 AY DP1 DP2 DP3 DP4
RESTRICT DP2:1 – DP1:2 = 0
RESTRICT DP3:1 – DP1:3 = 0
RESTRICT DP3:2 – DP2:3 = 0
RESTRICT DP1:1 + DP2:1 + DP3:1 + DP4:1 = 0
RESTRICT DP1:2 + DP2:2 + DP3:2 + DP4:2 = 0
RESTRICT DP1:3 + DP2:3 + DP3:3 + DP4:3 = 0
End
209

Rotterdam Model w it
*
Dq it
 b i
[
(without intercept term)
Dy t
  w it
*
Dp it
]
 i
 j c ij
Dp jt
  it
Dq it
 log


 q it q it

1



Dp jt
 log


 p jt p jt

1



Dy t
 log


 y t y t

1


 w
* it
 1
2
( w it
 w it

1
)
Symmetry c ij
 c ji
Homogeneity
 j c ij

0
 i
210

TEST DP2:1 – DP1:2 = 0
TEST DP3:1 – DP1:3 = 0
TEST DP3:2 – DP2:3 = 0
TEST DP1:1 + DP2:1 + DP3:1 + DP4:1 = 0
TEST DP1:2 + DP2:2 + DP3:2 + DP4:2 = 0
TEST DP1:3 + DP2:3 + DP3:3 + DP4:3 = 0
End
Possible to do Test for Slutsky Symmetry
Test for Homogeneity
Test for Additivity
Use Nonlinear Regression
NL 3 / NCOEF = 12 AUTO PCOV
CONV = .0001 ITER = 500
EQ W1DQ1 = B1*AY + C11*DP1 + C12*DP2 + C13*DP3 + C14*DP4
EQ W2DQ2 = B2*AY + C12*DP1 + C22*DP2 + C23*DP3 + C24*DP4
EQ W3DQ3 = B3*AY + C13*DP1 + C23*DP2 + C33*DP3 + C34*DP4
211

COEF B1 B2 B3
COEF C11 C12 C13 C14 C22 C23 C24
COEF C33 C34
End
But to insure Homogeneity
Replace C14 with –(C11 + C12 + C13)
Replace C24 with –(C12 + C22 + C23)
Replace C34 with –(C13 + C22 + C33)
Now NCOEF = 9
COEF B1 B2 B3
COEF C11 C12 C13
COEF C22 C23 C33
End
212

 * ij

 p j q i
 y
 q i
 y
 q j
 y k ij
Rotterdam Model
   y

 q y i

 q y j
  p j
  i
 q j
 y

 p j
  i
 j q j y
 j

 q j
 y y q j c ij
 w i
 * ij
  w i w j
 i
 j
Note that: b i
= w i
 i
 cij = -

[w i
 i
] [w i
 i
] = -
 b i b j
213

With Additivity
NL 3/NCOEF = AUTO PCOV CONV = .0001
ITER = 500
EQ W1DQ1 = B1*AY + C11*DP1 + (-PHI*B1*B2*DP2) + (-PHI*B1*B3*DP3)
+ (-PHI*B1*B4*DP4)
EQ W2DQ2 = B2*AY + (-PHI*B1*B2*DP1) + C22*DP2 + (-PHI*B3*B2*DP2)
+ (-PHI*B2*B4*DP4)
EQ W3DQ3 = B3*AY + (-PHI*B3*B1*DP1) + (-PHI*B3*B2*DP2) + C33*DP3
+ (-PHI*B3*B4*DP4)
COEF B1 B2 B3
COEF C11 C22 C33
B4 PHI
End
214

But need to impose Homogeneity
 j c ij

0

C11 + C12 + C13 + C14 = 0
C11 + (-

B1B2) + (-

B1B3) + (-

B1)[1-B1-B2-B3] = 0
B4 = 1-B1-B2-B3
C11 + (-

B1) +

B1B1 = 0
C11 =

B1B1 +

B1 = -

B1 [B1-1] =

B1 (1-B1)
Similarly C22 =

B2 (1 – B2)
C33 =

B3 (1 – B3)
215

Therefore, NCOEF = 4
COEF B1
End
B2 B3 PHI
Obtain log-likelihood function without additivity restriction
Obtain log-likelihood function with additivity restriction
LR Test:   max w
L / max
W
L with restrictions without restrictions ln
  ln

 max w
L

 ln

 max
W
L


-2 lnλ ~ X p
2 , where p = # restrictions to be tested
For additivity, p = 5
WHY? # independent parameters to be estimated under classical restrictions n
2
= , for n = 4
2 2

9
Under additivity 3 independent parameters to estimated

4 difference is 5
216

ln P t
 w it

0


 i

 k

 j k

AIDS Model ln ij ln p k p
 jt
1
2

B i ln
 k j


 kj y t
P t ln



  p k it ln p j ln
Stone’s Approximation
P t
  k w kt ln p kt
To avoid simultaneous bias, use w t-1 in lieu of w
Kt
.

Homogeneity
 ij

0
 i j

Symmetry ij
  ji
217

SHAZAM PROGRAM
For AIDS Model
/ restrict System 3 5
OLS W1
OLS W2
OLS W3
LOGP1 LOGP2 LOGP3
LOGP1 LOGP2 LOGP3
LOGP1 LOGP2 LOGP3
LOGP4 LOGYP
LOGP4 LOGYP
LOGP4 LOGYP
RESTRICT LOGP2:1 – LOGP1:2 = 0
RESTRICT LOGP3:1 – LOGP1:3 = 0
RESTRICT LOGP3:2 – LOGP2:3 = 0
RESTRICT LOGP1:1 + LOGP2:1 + LOGP3:1 + LOGP4:1 = 0
RESTRICT LOGP1:2 + LOGP2:2 + LOGP3:2 + LOGP4:2 = 0
RESTRICT LOGP1:3 + LOGP2:3 + LOGP3:3 + LOGP4:3 = 0
End
218

Additivity in the AIDS Model
 * ij
  w i
 i
 j
 i

1

B i w i
 w i

B i w i
 j

1

B w j j  w j

B j w j
 * ij
  w i



 w i
 w i
B i





 w j
 w j
B j
 
 w i

B i
  w j

B j




 * ij
 w i
For AIDS,

= -1
  * ij

( w i

B i
) ( w j

B j
) w i
219

Linkage of Demand Systems to Macroeconomics
The Extended Linear Expenditure System
The ELES is the traditional approach used to link demand systems to macroeconomic models. The ELES is derived from maximizing an intertemporal
Klein-Rubin utility function subject to a wealth constraint. The extension to the
LES is the addition of savings as an endogenous component of demand system.
The extended linear expenditure system is given as p i q i
 p i
 i
 
B i
( y
  p i
 i
), (1) where i = 1, …, n. The ELES expression consumption expenditure p function of prices p
1
, …, p n and income y under the restrictions i q i
0

B i

1 ;

B i

1; (and) p i q i
 p i
 i
 as a
0 .
(2)
Besides the assumptions of intertemporal additivity and static price expectations held with certainty, the ELES as given above incorporates the further assumption that the present value of expected changes in labor income are zero, so that permanent income and measured income are the same.
220

We obtain the aggregate consumption function associated with the ELES as
CE

 i p i q i

( 1
 
)
 i n p i
 i
  y .
(3)
Thus, the parameter
 in the ELES is the marginal propensity to consume (MPC). The above relationship is a Keynesian consumption function with the intercept defined as a linear function of prices. Given that consumption expenditure (CE) and savings (S) equal income (Y), then S = Y – CE. Savings then is equal to
S
 y ( 1
 
)

( 1
 
)
 i n p i q i

( 1
 
)[ y
  i n p i
 i
] (4)
Thus, 1-
 is the marginal propensity to save. The subsistence parameter associated with savings is zero. The ELES can be decomposed into the LES and the aggregate consumption function. Given the aggregate consumption function, we may also derive the aggregate savings function.
221

Income elasticities and uncompensated own-price and cross-price elasticities from the ELES are given as
Income Elasticity
 i



B w i i

 [
 
( 1
 
)

], (5)
Own-Price Elasticity
 ii



B w i i

 [

( 1
  i
)
  w i
], and (6)
Cross-Price Elasticity
 ij
 



B i w i


w j

1


B j w j



, (7) respectfully, where
 is the supernumerary ratio

[
CE


CE p i
 i
].
(8)
222

The ELES system is important because of the inclusion of savings into a demand system. In essence, the ELES creates a residual category of savings, and consumers allocate a measure of income or wealth to expenditures of goods and services and savings. Savings is not treated as an expenditure category. As such, with the ELES, while it is possible to estimate the MPC and
MPS, it is not possible to estimate the own-price elasticity of savings. Indeed, this situation is a shortcoming of the ELES. It is possible, however, to estimate the effects of changes in prices and income on savings. The effect of price changes on savings has been largely unexplored. These effects can be derived by taking the derivatives of S with respect to y and p i in equation (4).
223

Data Issues
Only households that could be tracked for all four quarters of the 1989
BLS Consumer Expenditure Survey were used in this analysis. The expenditure categories are the following: (1) food at home; (2) food away from home; (3) housing (less utilities); (4) apparel; (5) transportation (less gas and oil); (6) energy; (7) health care; and (8) miscellaneous expenditure. The energy category consists of expenditures for utilities, gasoline, and motor oil. The miscellaneous expenditure category is the difference between total consumption expenditure and the sum of the first seven categories. Savings is the residual between income
(after taxes) and total expenditure.
Annual expenditures were calculated by summing over the four quarters.
All observations corresponding to expenditures or income levels lower than the first percentile or greater than the 99 th percentile of the sample were eliminated from the analysis to reduce any problems with outliers; as well, negative incomes and expenditure values were omitted. Finally, with this cross-sectional data set, it is possible for savings to be negative due to the fact that total expenditures may exceed after-tax income. However, households whose total expenditures were twice as great as after-tax income were eliminated from the analysis. The number of households meeting all relevant criteria, and consequently the number of households used in this analysis, was 491.
224

Theoretical Model
One of the most popular demand systems for estimating aggregate commodity expenditures has been the venerable linear expenditure system (LES) first introduced by Stone in 1954. In fact, the LES is the foundation for the development of the ELES (Lluch; Howe). However, a principal drawback of the
LES is linear Engel functions. The quadratic expenditure system (QES), introduced by Howe, Pollak, and Wales, is a generalization of the linear expenditure system in which the Engel functions are no longer linear, and consequently, the marginal budget shares vary with the level of total expenditure, income, or wealth.
Howe, Pollak, and Wales show that all demand equations that are quadratic in expenditure are generated by the indirect utility function

(P,y)

-g(P) y-f(P)
 h(P) g(P)
. (1)
Y corresponds to total expenditure, P is a vector of prices, and g(P), h(P), and f(P) are functions linearly homogeneous in P.
225

Theoretical Model, cont.
We employ the same specification used by Howe, Pollak and Wales, and by Barnes and Gillingham in which g(P)
  p a k k
, f ( P )

 p k b k
, h ( P )

 p k c k
, and
 a k

1 , (2)
Where a, b and c are parameters of the QES, and k is the number of commodity groups or expenditure categories. By combining equations 1 and 2, the indirect utility function is defined as

(P,y)
 y-

 p k a p k k b k


 p k p k a k c k
. (3)
By employing Roy’s identity, the demand functions (in expenditure form) are
E i expressed as follows
 p i b i
 a i
( y

 p k b k
)

( c i p i
 a i
 p k c k
)
 p

2 a k k
( y

 p k b k
)
2
.
( 4 )
This specification of the QES approaches the LES as c i approaches 0 for all i.
226

In conventional demand analyses, y corresponds to total expenditure. In this analysis, we also show y to refer to wealth. We define wealth, due to restrictions on available data, as income after taxes, available lines of credit less outstanding balances, savings, and other financial assets. With this definition, we construct a demand system wherein debt repayment/saving is a separate expenditure category. With this representation, we are able to consider the allocation of wealth, or perhaps better stated, total available funds among n consumption categories in this analysis are: (1) food at home; (2) food away from home; (3) housing (less utilities); (4) apparel; (5) transportation (less gas and oil);
(6) energy; (7) health care; (8) miscellaneous expenditure, and (9) debt repayment/saving. The energy category consists of expenditures for utilities, gasoline, and motor oil. The miscellaneous expenditure category is the difference between total expenditure on consumption categories and the sum of the first seven categories. Debt repayment/saving is the residual between wealth (total available funds) and total expenditure.
Data
There is no single data set that includes information on expenditures, prices, financial assets, and demographics for individual households. We combine several data sets to create this “master” cross-sectional data set. In this section, we discuss these various data sets.
Income and expenditure data come from the interview portion of the
Bureau of Labor Statistics (BLS) Consumer Expenditure Survey (CES) for 1989.
227

BARTEN - Classes of Differential Demand Functions
Rotterdam:
AIDS: w i d dw i ln
 q i
T i d
T i
 b i

 ln w i b i d
Q
 ln Q

 j
 ij d
 j c ij d ln p j ln p j
 ij
 c ij
 w i
 ij
 w i w j
CBS (Central Bureau of Statistics)
Keller & Van Driel (1985) w i
( d ln q i
 d ln Q )
 b i d ln Q

 j c ij d ln p j
This system has the AIDS income coefficients and the Rotterdam price coefficients.
NBR (Neves (1987)) dw i
 w i d ln Q
 b i d ln Q


 ij d ln p j j
This system has the Rotterdam income coefficients and the AIDS price coefficients.
228

For purposes of estimation, replace the differentials by finite first differences and the w i w
 1
2
( w
 w can add an intercept to represent trends.
).
General Model of Barten: w i d ln q i
 b i d ln Q

 j c ij d ln p j
 
1
( w i d ln Q )
 
2
( w i d ln q i
 dw i
 w i d ln Q )
If:

1

1

1

1
=

2
=

2
= 0
= 1
= 1;

2
= 0;

2


= 0
= 1
Rotterdam
AIDS


CBS
NBR
229